Here’s this week’s Sunday Function. In the universe of functions it’s an utterly typical suburban middle class citizen, with a pleasant but quite ordinary job in a downtown cubicle farm for Physics Incorporated.
His name is
In a little more detail, you’ll notice that this function is the product of two elementary functions. There’s x squared, and there’s e raised to the negative x power. As x increases, obviously x squared increases quite quickly. By the time you get to x = 500, x squared is equal to 250,000. But on the other hand, e^-x is shrinking as x increases. It happens to be shrinking even faster than x squared is growing. By the time x = 500, e^-x is equal to approximately a mere 0.000…00712, where there’s a total of 217 zeros to the right of the decimal point. The shrinking exponential “wins” and as x gets large, f(x) gets small.
This is a reflection of a more general mathematical truth. If we has x cubed instead of x squared, the function would still go to zero as x increased. The negative exponential still wins. In fact, any constant power of x is going to be “beaten” by the exponential, which grows faster than any power of x.
These kinds of relationships are very important in physics. It may be very hard to pry apart the specifics of some complicated function, but if you recognize which parts dominate the function for large x, you may be making some very important headway. The general name for this type of examination goes by the rather suggestive moniker of Big O notation.
In physics we don’t usually come across functions which grow faster than the negative exponential shrinks, but they do exist. The factorial function is one of the most commonly encountered.
Pure mathematics has functions which just grow incomprehensibly big at an astonishing rate. Here’s an excellent and very readable essay on the subject of large numbers by Scott Aaronson.
It’s a nice way to realize that even the simple tools of math can quickly be extended into some truly arcane territory.